I need to find the principal part of the Laurent series for $f(z) = \frac{e^{2z}}{1-\cos(z)}$, around $z = 0$. Also, I have to use the undetermined coefficient method.
I don't know how to proceed. Should I write down the series for each part and then divide it, this is probably the long way? What is this method in the instruction?
The "undetermined coefficient method" is probably the method where you make an ansatz
$$\frac{f(z)}{g(z)} = \sum_{n=-k}^\infty a_n z^n$$
and determine the coefficients $a_n$ by computing the Cauchy product
$$\sum_{r=0}^\infty \varphi_r z^r = \left(\sum_{n=-k}^\infty a_n z^n\right)\cdot \left(\sum_{m=0}^\infty \gamma_m z^m\right),$$
where the $\varphi_r$ and $\gamma_m$ are the coefficients of the Taylor series of $f$ resp. $g$,
$$f(z) = \sum_{r=0}^\infty \varphi_r z^r;\qquad g(z) = \sum_{m=0}^\infty \gamma_m z^m.$$
In this ansatz, $k$ is the order of the pole, it does not work for essential singularities.
Since $k$ is small here, and you are only interested in the principal part, it is not much work.